Elastic constitutive laws for incompatible crystalline media: the contributions of dislocations, disclinations and G-disclinations

Abstract

Linear higher-grade higher-order elastic constitutive laws for compatible (defect-free) and incompatible (containing crystal line defects) media are presented. In the proposed model, the free energy density of a body subjected to elastic deformation under the action of surface tractions, moments or hyper-traction tensors (second-order tensors whose anti-symmetric part corresponds to moments) has contributions coming from the first two gradients of displacements. Thermodynamic considerations reveal that only the symmetric component of the gradient of elastic displacement, i.e., compatible elastic strain tensor, and the anti-symmetric component of the second gradient of elastic displacement, i.e., compatible third-order elastic curvature tensor, contribute to the free energy density during compatible deformation of the body. The line crystal defect contributions are accounted for by incorporating the incompatible components of elastic strains, curvatures and symmetric 2-distortions as state variables of the free energy density. In particular, the presence of generalized disclinations (G-disclinations) is acknowledged when the medium is subjected to surface hyper-traction tensors having a non-zero symmetric component along with surface-tractions on its boundary. Mechanical dissipation analysis provides for the coupling between the Cauchy stresses and third-order symmetric hyper-stresses. The free energy density and elastic laws for a defect-free and line crystal defected medium are proposed in a linear setting. In the special case of isotropy, the cross terms between elastic strains and curvatures contribute to the free energy density through a single elastic constant. More interestingly, the Cauchy and couple stresses are found to have contributions coming from both, elastic strains and curvatures.